Momentum, impulse and collisions

conservation of momentumimpulse momentum theoremelastic and inelastic collisionscoefficient of restitution

Momentum and impulse, the conservation law that links two bodies, and the elastic, inelastic and restitution results for what happens after they meet.

Linear Momentum (p = mv)

p = m v

Momentum as the product of an object's mass and velocity.

Impulse (J = FΔt)

J = F \, \Delta t

Impulse delivered by an average force acting over a contact time, equal to the change in momentum.

Conservation of Momentum (Two Bodies)

m_1 u_1 + m_2 u_2 = m_1 v_1 + m_2 v_2

Conservation of linear momentum in a two-body collision, solving any one mass or velocity from the other five.

Perfectly Inelastic Collision

v = \frac{m_1 u_1 + m_2 u_2}{m_1 + m_2}

Common velocity of two bodies that stick together after a perfectly inelastic collision, from conservation of momentum.

Elastic Collision — Final Velocity of Body 1

v_1 = \frac{\left(m_1 - m_2\right) u_1 + 2 m_2 u_2}{m_1 + m_2}

Final velocity of the first body in a one-dimensional elastic collision, where both momentum and kinetic energy survive.

Coefficient of Restitution

e = \frac{v_2 - v_1}{u_1 - u_2}

Ratio of separation speed to approach speed in a collision, measuring how much of the relative motion survives impact.

Bounce Height from Coefficient of Restitution

h_2 = e^{2} h_1

Height a dropped ball rebounds to, from the drop height and the coefficient of restitution of the bounce.

How they fit together

Momentum is conserved in every collision, full stop — no exceptions, no caveats about the type of impact. Impulse is the same statement rearranged: a force applied for a time changes momentum by exactly F Δt, which is the whole design principle behind crumple zones and airbags. They do not reduce the momentum change, which is fixed by how fast you were going; they stretch the time so the force comes down.

What separates the collision formulas is kinetic energy, not momentum. Perfectly inelastic means the bodies stick together and share one final velocity — use it whenever the question says 'embeds', 'couples' or 'they move off together'. Elastic means kinetic energy is also conserved, which gives a second equation and so a unique pair of final velocities; it is a good model for billiard balls and gas molecules and a poor one for cars. Everything in between needs the coefficient of restitution, measured from a drop test: e is the ratio of separation speed to approach speed, 1 for perfectly elastic and 0 for perfectly inelastic. The persistent mistake is assuming energy is conserved because momentum is. In a real crash most of the kinetic energy is gone, into bent metal and heat, while the momentum is untouched.